Poor Pigs

Problem Statement

In a scenario where you are given a number of buckets (each filled with liquid), and exactly one of those buckets contains a poisonous substance, the objective is to efficiently determine which one is toxic. You can use a group of pigs to test the buckets by feeding them the liquids, observing which pigs die, and thereby narrowing down or identifying the poisonous bucket. The real challenge lies in the constraints given: you have a limited amount of time (specified by minutesToTest) during which to conduct your experiments, and it takes a certain amount of time (minutesToDie) after a pig has consumed the poisonous substance for it to die.

This problem requires a strategic approach to minimize the number of pigs used while ensuring you can conclusively identify the toxic bucket within the allowed testing time. Each pig can test multiple buckets at once by drinking from them simultaneously, and the combination of which pigs die after a round of testing gives you information about which specific bucket or set of buckets could be poisonous.

Examples

Example 1

Input:

buckets = 4, minutesToDie = 15, minutesToTest = 15

Output:

2

Explanation:

We can determine the poisonous bucket as follows:
At time 0, feed the first pig buckets 1 and 2, and feed the second pig buckets 2 and 3.
At time 15, there are 4 possible outcomes:
- If only the first pig dies, then bucket 1 must be poisonous.
- If only the second pig dies, then bucket 3 must be poisonous.
- If both pigs die, then bucket 2 must be poisonous.
- If neither pig dies, then bucket 4 must be poisonous.

Example 2

Input:

buckets = 4, minutesToDie = 15, minutesToTest = 30

Output:

2

Explanation:

We can determine the poisonous bucket as follows:
At time 0, feed the first pig bucket 1, and feed the second pig bucket 2.
At time 15, there are 2 possible outcomes:
- If either pig dies, then the poisonous bucket is the one it was fed.
- If neither pig dies, then feed the first pig bucket 3, and feed the second pig bucket 4.
At time 30, one of the two pigs must die, and the poisonous bucket is the one it was fed.

Constraints

• 1 <= buckets <= 1000
• 1 <= minutesToDie <= minutesToTest <= 100

Approach and Intuition

Understanding the underlying logic of this problem can be simplified by breaking it down into steps and considering some key observations:

1. Each pig can provide information on the buckets it tested based on whether it survives or dies after minutesToDie. The testing strategy can be designed such that you are encoding which bucket could potentially be poisonous based on these outcomes.

2. The pigs can be tested in rounds, with the total number of rounds determined by minutesToDie and minutesToTest. That is, you can only run a limited number of tests based on these two parameters, where each test lasts for minutesToDie and the total testing duration cannot exceed minutesToTest.

3. Optimizing the number of pigs involves using a combination approach:

   • Use the first round to test a set arrangement of buckets with designated pigs. Based on which pigs die, use successive rounds to focus on fewer buckets with fewer or the same number of pigs until the poisonous bucket is identified or till all rounds are exhausted.

4. To elucidate with examples:

   • Example 1: Given 4 buckets and only one round of testing allowed (minutesToTest equals minutesToDie equals 15 minutes):

     • Feed Pig 1 with buckets 1 and 2.
     • Feed Pig 2 with buckets 2 and 3.
     • After 15 minutes, the combination of which pig dies points directly to the poisonous bucket. This method is highly efficient given the constraints, using only 2 pigs.

   • Example 2: Given 4 buckets, but now you have two rounds of testing (minutesToDie is 15 minutes but minutesToTest is 30 minutes):

     • First test:
       • Feed Pig 1 bucket 1, and Pig 2 bucket 2.
     • Depending on the results, in the second round:
       • If neither dies, test Pig 1 with bucket 3 and Pig 2 with bucket 4.
     • The death of the pig in either round pinpoints the poisonous bucket.

   This approach leverages the binary logic enabled by pig's survival or death to iteratively halve the set of possible poisonous buckets each round, leading to a solution with a minimum number of pigs used effectively within the allotted test time.

Solutions

class Solution {
  public:
  int calculatePigs(int buckets, int deathTime, int totalTime) {
    int rounds = totalTime / deathTime + 1;
    return ceil(log2(buckets) / log2(rounds));
  }
};

class Solution {
  public int calculatePigs(int totalBuckets, int timeToDie, int totalTime) {
    int possibleStates = totalTime / timeToDie + 1;
    
    return (int) Math.ceil(Math.log(totalBuckets) / Math.log(possibleStates) - 1e-10);
  }
}